An orbital strategy for regulating the Jahn–Teller effect

ABSTRACT The Jahn–Teller effect (JTE) arising from lattice–electron coupling is a fascinating phenomenon that profoundly affects important physical properties in a number of transition-metal compounds. Controlling JT distortions and their corresponding electronic structures is highly desirable to tailor the functionalities of materials. Here, we propose a local coordinate strategy to regulate the JTE through quantifying occupancy in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{d}_{{{z}^2}}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{d}_{{{x}^2} - {{y}^2}}}$\end{document} orbitals of Mn and scrutinizing the symmetries of the ligand oxygen atoms in MnO6 octahedra in LiMn2O4 and Li0.5Mn2O4. The effectiveness of such a strategy has been demonstrated by constructing P2-type NaLixMn1–xO2 oxides with different Li/Mn ordering schemes. In addition, this strategy is also tenable for most 3d transition-metal compounds in spinel and perovskite frameworks, indicating the universality of local coordinate strategy and the tunability of the lattice–orbital coupling in transition-metal oxides. This work demonstrates a useful strategy to regulate JT distortion and provides useful guidelines for future design of functional materials with specific physical properties.

Lindeman glass capillary with an internal diameter of 0.3 mm.Synchrotron powder X-ray profiles were measured at a SPring-8 BL19B2 beamline.A large Debye-Sherrer camera with an imaging plate detector was used for data collection.Data was collected at room temperature.The wavelength of the incident Xrays was 0.41334 Å, as calibrated by an NIST CeO 2 standard sample.All data were collected with sin θ/λmax = 1.67 Å −1 .
Before performing the multipole refinement, accurate structure factors are needed.Quantitative CBED was implemented to accurately measure the low-order crystal structure factors that are sensitive to valence electrons, as illustrated in Fig. S9, where structure factor measurements were made by comparing experimental intensity profiles across QCBED disks (rocking curves) with calculations, using a goodnessof-fit (GOF) criterion 1 .Tables S1 and S2 show the refined and converted structure factors.Considering that low temperature can induce phase transition of the spinel LiMn 2 O 4 , our QCBED experiments were performed at 300K (room temperature).Since battery materials are highly sensitive to electron beam damage, we carefully optimized the experimental conditions for QCBED to avoid the beam damage.Both the CBED patterns and electron energy loss spectra indicate that the electron dose used in our experiment did not induce atomic and electronic structural changes (see Figs. S10-11, Table S4).Since the high-order structure factors are primarily determined by the atomic position and core electrons, X-ray diffraction or DFT calculations can be employed to complement the higher-order structure factors without losing significant accuracy.In this study, the higher-order structure factors were calculated using WIEN2K 2 with a fully-potential linear augmented plane-wave method 3 , that can guarantee highly accurate results.After acquiring the QCBED pattern of Li 1-x Mn 2 O 4 , we used a to refine the low-order structure factors 1 .The refined parameters include the electron-beam incident direction, the sample thickness, and the structure factors.To avoid very large matrices and a reasonable accuracy for the Bloch wave theory, electron beams were selected according to certain criteria 4 .2.6 for g max , 3.0 for 2KS gmax , 0.005 for |   2  | were adopted for refinement, respectively.
Multipole refinement provides the most efficient parameterization of the real-space charge density and gives a result that is insensitive to missing reflections in the collected dataset 5,6 .In this method, the crystal charge density is fitted by a sum of non-spherical pseudo-atomic densities.With the refined population parameters for real spherical harmonics in the valence part, one can determine the charge transfer and rearrangement of orbital due to the chemical bonding and the local symmetries 5,7 .To perform the multipole refinement, we need to convert the refined electron structure factors to the X-ray structure factors through the Mott-Bethe formula.The refined low-order electron and converted X-ray structure factors are listed in Table S1-S2 for LiMn 2 O 4 and Li 0.5 Mn 2 O 4 , respectively.In this work, the high-order structure factors are obtained from the DFT calculations using the full potential linear augmented plane wave method implemented in WIEN2K 2 .
The multipole refinements were performed in JANA2006 software 8 .As shown in Eq.S1, the multipole model is based on the real spherical harmonics which centered around each atomic nucleus 5,9 .As to d electrons, the l max = 4 at the hexadecapole level is enough.The coordinate systems used for both Mn (3 ̅ ) and O (3) are that the z-axis is parallel to the 3-fold axis [111] direction, and the y axis is perpendicular to the mirror plane, i.e., alone [11 ̅ 0] direction.Several constraints were imposed on the multipole refinement: the valence population of Li ion was fixed at P v = 0, κ [Li] = 1 and the deformation electron density of Li was not refined.For Mn, the two 4s electrons were treated as core electrons, and the five 3d electrons were considered as valence electrons 10 .
is the valence electron population parameter. ± is real spherical harmonics with the population parameter  ± .The  and  ′ parameters characterize the radial expansion-contraction of the valence electron density for the spherical and aspherical part, respectively.  is the radial function.

Electron dose and beam damage
Cathode materials are very sensitive to the electron beam.To avoid and reduce the beam damage, when collecting the diffraction patterns, we continually keep moving the sample and grasping as fast as possible to reduce the recording time.Due to the different d-spacings, we collected the CBED data with two different condenser lens (CL) apertures while keeping other conditions unchanged.For (111) and ( 222), the convergent semi-angle of 3.1 mrad was adapted.For other three structure factors, the convergent semiangle of 4.3 mrad was adapted.All the experimental conditions were the same for LiMn 2 O 4 and Li 0.5 Mn 2 O 4 , including the convergent semi-angles.Table S4 presents the beam currents and electron doses of the above mentioned two experimental conditions.
As the beam damage could induce the change of atomic and electronic structures, we performed electron diffraction and electron energy loss spectroscopy to verify that the dose used in the QCBED experiments is under the safe range.In general, compared with parallel beam electron diffraction, the diffraction discs of CBED contain rich structural information, and therefore are more sensitive to structural changes 11 .Fig. S10 exhibits some CBED patterns of LiMn 2 O 4 , which clearly show the changes of the contrast before and after beam damage.We selected the CBED patterns without any fringes and HOLZ line splitting during refining the structure factors.In addition, due to the high sensitivity of the CBED pattern to structural changes, defects induced by beam damage will lead to a large deviation between the refinement results and the experimental intensity.
Besides, STEM-EELS experiments of the specimen were performed to detect the electronic structure changes.The electron doses used in CBED collection were too small to obtain adequate signal to noise ratio in TEM-EELS.Thus, we performed STEM-EELS to evaluate the electronic structure changes during the electron beam irradiation.The beam current is 120 pA, which is equal to the largest beam current used in CBED.To simulate the situations in TEM mode, we set a square area with 0.308×0.308nm, and adjust the defocus slightly.In this case, the electron beam did not focus on the sample, and has almost the same diameter with the irradiated area.The pixels of the selected irradiated area were divided into 30×30, and the exposure time was 0.01 s.Each 10 spectra were superposed to obtain an EELS spectrum collected with 0.1 s.Fig. S11 illustrates the EELS spectra of LiMn 2 O 4 and Li 0.5 Mn 2 O 4 and shows that the electronic structures start to change slightly at 0.3 s for both samples, indicating that the electron dose used in CBED collection (exposure time of 0.1 s) would not induce the obvious electronic-structure changes of the samples.

DFT calculations
The In both cases, the PBE-GGA was adopted as the exchange-correlation functionals, and the select energy and the energy convergence criterion were set to be -6.0Ry and 10 -5 eV respectively.
To consolidate the reliability of our mechanism, we constructed several configurations of manganesebased oxide cathodes, spinel-and perovskite-type 3d transition-metal oxides with different coordinated oxygen symmetries, and analyzed the octahedron distortion and electronic structures of the corresponding M (from Ti to Zn) atoms.In order to regulate the coordinated oxygen symmetries, we select spinel of the M atoms by the Li atom to construct the (2, 4)-MO 6 octahedrons.After constructing the structures, we calculate the electronic structures within the GGA+U scheme, with the projector augmented wave method 12 as implemented in the VASP 13,14 .The atom positions are relaxed with the conjugated gradient method, and the final force on each atom is less than 0.005 eV/Å., which is based on the bond length 28  , which is based on the bond length 28 .  is the average bond length and   is the bond length from the ith ligand atom to the center transition-metal atoms in the MX 6 octahedron.

Supplementary Note 1 .
Mapping the electron density and d-orbital populations QCBED collection.The QCBED experiments were performed using a FEI Tecnai G2 F20 S-TWIN transmission electron microscope equipped with a Gatan imaging filter and 1024 × 1024-pixel chargecoupled device camera.Near the zero-loss peak, select an energy window of 10 eV.Due to the use of energy filters in QCBED experiments, inelastic scattered electrons are excluded from the Bloch wave calculations.Two convergence angles were used to obtain the systematic row patterns depending on the different d-spacings of the sample.The accelerating voltage was precisely measured at 198.50 kV through fitting the QCBED pattern from dynamic simulations with experimental patterns obtained from singlecrystal silicon samples.SPXRD measurements.Polycrystalline samples of LiMn 2 O 4 and Li 0.5 Mn 2 O 4 were sealed in a

LiMn 2 O 4 ,
O3-layered LiMnO 2 and O1-layered LiMnO 2 , they share corners, edges and faces with the adjacent LiO x polyhedron, respectively.Removing the Li atoms from the above structures leads to form the (6, 0),(3, 3), and (2, 4) configurations in these structures.For the spinel Li 1-x M 2 O 4 framework, the constructed structure is the same as the Li 1-x Mn 2 O 4 .For the perovskite LaMO 3 framework, we replace 1/6

Fig. S6 .
Fig. S6.Expanded synchrotron powder X-ray diffraction pattern for Li 0.5 Mn 2 O 4 .No splitting of diffraction peaks indicates there is no phase transition from the cubic to tetragonal structure.

Fig. S10 .
Fig. S10.Comparison of CBED patterns with and without structural changes.(A-D) are (004) systematic row CBED patterns of LiMn 2 O 4 .(A) is the diffraction pattern without structure change.The messy contrast and crooked rocking curves in (B), and the sideling fringes in (C-D) indicate the structure changes during beam radiation.

Fig. S11 .
Fig. S11.EELS spectra of the (A) LiMn 2 O 4 and (B) Li 0.5 Mn 2 O 4 .Electronic structures changed at 0.3s as indicated by the green triangle.

Fig. S12 .
Fig. S12.Experimental static deformation electron density maps of the (001) MnO 4 plane of (A) LiMn 2 O 4 and (B) Li 0.5 Mn 2 O 4 .The contour interval is 0.1  Å −3 , with positive and negative contours drawn as solid red and dashed blue lines, respectively.(C-D) The corresponding difference electron density map between the calculated and the neutral atoms of LiMn 2 O 4 and Li 0.5 Mn 2 O 4 , respectively.
Vienna Ab Initio Simulation Package (VASP) based on the DFT was used to calculate the electron density and electronic structures of LiMn 2 O 4 , Li 0.5 Mn 2 O 4 , layered-LiMnO 2 , layered-Li 0.67 MnO 2 , layered-Li 0.5 MnO 2 , spinel Li 1-x MO 2 and perovskite La(Li x M 1-x )O 3 .The Perdew-Burke-Ernzerhof (PBE)functional within a generalized gradient approximation (GGA) form was adopted to treat the exchangecorrelation energy.DFT+U was used to correct the self-interaction error of conventional DFT for correlated d electrons.U values of 3.25, 3.7, 3.9, 5.3, 3.32, and 6.2 eV were used for V, Cr, Mn, Fe, Co, and Ni, respectively.A plane wave representation for the wave function with a cut off energy of 500 eV was applied.Geometric optimizations were performed using conjugate gradient minimization until all forces acting on the ions were less than 0.01 eV/Å per atom.The K-point mesh with a spacing of ca.0.03 Å −1 was adopted.The extended supercells of spinel-Li 8 M 16 O 32 , layered-Li 12 Mn 12 O 24 , and perovskite- La 12 (Li 2 M 10 )O 36 were adopted for the geometric optimizations and electronic structures (where M indicates a 3d TM).Beside DFT+U, we also adopted HSE06 on LiMn 2 O 4 and Li 0.5 Mn 2 O 4 with the standard mixing parameter (α = 0.25) to verify the results from DFT+U calculation and the experiment.WIEN2k using a fully-potential linear augmented plane-wave method was adopted to obtain the structure factors of LiMn 2 O 4 and Li 0.5 Mn 2 O 4 via Fourier transformation of the theoretical electron density.The input structure of LiMn 2 O 4 was set as refined from experimental data, while the optimized structure from VASP was used for Li 0.5 Mn 2 O 4 .

Table S11 .
.   is the average bond length and   is the bond length from the ith ligand atom to the center transition-metal atoms in the MX 6 octahedron.Structures containing the (2, 4) configuration of MX 6 octahedra, where there is JT distortion.